2007-06-22 08:18:42 · 9 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
Answer: 4
This can be done with or without a calculator.
Calculator method:
Punch it into a calculator and see what the remainder is.
Without calculator:
Investigate power of 4 and the remainder when it is divided by 10.
4^1 = 4 (the remainder is 4)
4^2 = 16 (the remainder is 6)
4^3 = 64 (the remainder is 4)
4^4 = 256 (the remainder is 6)
This pattern continues. If the power is odd, then you will get a remainder of 4.
2007-06-22 08:22:30 · answer #1 · answered by MsMath 7 · 1⤊ 1⤋
Well, the answer needs to be proved mathematically. Here'S the proof.
4^1 = 4
4^2 = 4*4 = 16
4^3 = 4 *4*4 = 84
All multiples of ten need to end in a zero. 4^2 = 16 and the remainder is 6. Therefore 4^3 will end in 4 as 6 * 4 = 24 and the unit digit stays in the unit digit without any addition. When 4^3 , ending in 4, is multiplied by 4, the unit digit of 4^4 will be 6 as 4 *4 =16
Thus thid alternating pattern will continue with every odd exponent resulting in a numer ending in 4, and every even in 6.
Therefore the remainder of 4^31 =4
2007-06-22 15:58:44 · answer #2 · answered by mint 2 · 1⤊ 0⤋
If n>0 is odd, then the remainder when 4^n is divided by 10 is 4. If n is even, the remainder is 6. To prove this, observe that if n=1, then 4^1 = 4 and 4^1 = 4 (mod 10) (here, = means congruent to). So, the remainder of the division of 4^1 by 10 is 4 and the proposition holds for n=1.
Now, suppose it holds for some odd n. Then, 4^n = 4 (mod 10). The next odd number is n+2. According to the properties of congruences, we have
4^2 * 4^n = 4^2 * 4 (mod 10) => 4^(n +2) = 64 (mod 10). Since 64 - 4 = 60 is divisible by 10, it follows 64 = 4 (mod 10), which implies by transitivity that 4^(n +2) = 4 (mod 10), that is, the remainder of the division of 4^(n +2) by 10 is 4. This completes the induction and shows our proposition is true. Since 31 is odd, the answer is 4.
The proof for even n is similar, but here n is odd.
2007-06-22 15:52:48 · answer #3 · answered by Steiner 7 · 1⤊ 0⤋
4
2007-06-22 16:59:47 · answer #4 · answered by UNIQUE 3 · 0⤊ 1⤋
4^1= 4
4^2= 16
4^3 =64
4^4= 256
4^5= 1024
...
After this point, last digit is 4 or 6. If you count, for odd powers it is 4.
Since 31 is an odd power, the remainder will be 4.
2007-06-22 15:25:25 · answer #5 · answered by Leprechaun 6 · 1⤊ 0⤋
4611686018427387904 = 4^31
meaning 4 is the remainder
2007-06-22 15:25:16 · answer #6 · answered by Raccoon 3 · 1⤊ 0⤋
4^31 = 4.611.686.018.427.387.904
So the answer is 4 since that is the last digit
2007-06-22 15:24:30 · answer #7 · answered by Voice of Insanity 5 · 1⤊ 0⤋
4; last digit of powers of 4 alternate between 6 (even powers) and 4 (odd).
2007-06-22 15:24:48 · answer #8 · answered by John V 6 · 1⤊ 0⤋
I like math.... ummm 4 i think
2007-06-22 15:26:07 · answer #9 · answered by SAP 1 · 1⤊ 0⤋